INFINITARY GENERALIZATIONS OF DELIGNE’S COMPLETENESS THEOREM Christian Esp´ındola Abstract Given a regular cardinal κ such that κ<κ = κ (or any regular κ if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the κ-separable toposes. These are equivalent to sheaf toposes over a site with κ- small limits that has at most κ many objects and morphisms, the (basis for the) topology being generated by at most κ many covering families, and that satisfy a further exactness property T . We prove that these toposes have enough κ-points, that is, points whose inverse image preserve all κ-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when κ = ω, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call κ-geometric, where conjunctions of less than κ formulas and existential quantification on less than κ many variables is allowed. We prove that κ-geometric theories have a κ-classifying topos having property T , the universal property being that models of the theory in a Grothendieck topos with property T correspond to κ-geometric morphisms (geometric morphisms the inverse image of which preserves all κ-small limits) into that topos. Moreover, we prove that κ-separable toposes occur as the κ-classifying toposes of κ-geometric theories of at most κ many axioms in canonical form, and that every such κ-classifying topos is κ-separable. Finally, we consider the case when κ is weakly compact and study the κ-classifying topos of a κ-coherent theory (with at most κ many axioms), that is, a theory where only disjunction of less than κ formulas are allowed, obtaining a version of Deligne’s theorem for κ-coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic. Keywords : classifying topos, infinitary logics, completeness theorems, sheaf models. arXiv:1709.01967v1 [math.LO] 6 Sep 2017 1 Introduction This paper is a continuation of the investigation begun in [Esp17] on infinitary categor- ical logic, focusing now on infinitary generalizations of Deligne’s completeness theorem. This theorem asserts that a coherent topos has enough points, which is essentially G¨odel completeness theorem for finitary first-order classical logic. Makkai and Reyes in [MR77] prove that the same is true for the so called separable toposes, those toposes of sheaves on a site that has countably many objects and morphisms and whose topol- ogy is generated by countably many covering families. This result is in turn related to the completeness of countably axiomatized theories in Lω1,ω. It turns out, as we prove in the present paper, that this result can be generalized in a way that ω is replaced with any regular cardinal κ such that κ<κ = κ (a condition satisfied for inaccessible 1 κ, or, under the Generalized Continuum Hypothesis, for any regular κ), and that, nat- urally, the result is essentially a completeness theorem for what we call κ-geometric logic. This logic is an extension of geometric logic in which arities of function and relation symbols are cardinals less than κ, and one can take conjunctions of less than κ many formulas and existential quantification of less than κ many variables. The the- ory of well-orderings, for instance (see [Dic75]), cannot be expressed in finite-quantifier languages, but it is κ-geometric. In this more expressive extension there are valid sequents that cannot be derived from the usual axioms of geometric logic even if one extends the usual rules and axioms for conjunction and existential quantification. For example, the axiom of choice is expressible in the form: ∃xiφ(xi) ⊢x ∃i<γxi φ(xi) i<γ i<γ ^ ^ and is certainly valid (in all Set-valued models) but not derivable. However, this more expressive extension valid sequents can be made derivable in κ- geometric logic with the addition of one special rule, which is a refined version of transfinite the rule of transitivity introduced in [Esp17]. This rule corresponds to an exactness condition T satisfied by κ-geometric categories (the categories associated to κ-geometric theories), and it turns out that κ-geometric theories have what we call a κ-classifying topos. This is a Grothendieck topos satisfying the exactness condition T , where there is a generic model of the theory whose image along the inverse image of κ-geometric morphisms (those geometric morphisms whose inverse image preserve κ-small limits) corresponds precisely to the models of the theory in any other topos that satisfies the same exactness condition T . In the same way that countably axiomatized geometric theories are complete (with respect to Set-valued models), one can prove that κ-geometric theories with at most κ many axioms are also complete. The restriction on the cardinality of the axiomatization will then imply that the corresponding κ-classifying topos will be κ-separable, and by completeness it will be possible to prove that it has enough κ-points (points whose inverse image preserve κ-small limits). Since property T is valid in Set, the existence of enough κ-points for a given topos necessarily implies that the topos has property T , whence its naturalness in the definition of κ-separable toposes. Indeed, a localic topos Sh(L) without points cannot be κ-separable for κ> 2L, even if its site does have κ-small limits, at most κ many objects and morphisms, and the (basis for the) topology is generated by at most κ many covering families. In the particular case when κ is a weakly compact cardinal, the exactness property T on the κ-classifying topos of a κ-coherent theory (a κ-geometric theory where all disjunctions in the axioms are indexed by ordinals less than κ) that is axiomatized with at most κ-many axioms, adopts the form of the transfinite transitivity rule of [Esp17]. In this case, the topos is equivalent to sheaves on a site where Grothendieck topology is generated by families of less than κ many morphisms, and its κ-separability, that implies that it has enough κ-points, is the precise generalization of Deligne’s completeness theorem from coherent toposes to κ-coherent toposes. It turns out that the corresponding completeness theorem for κ-coherent theories (as- 2 sociated with Deligne’s theorem for κ-coherent toposes) adopts, via Morleyization, the form of Karp’s completeness theorem for Lκ,κ, the exactness property T corresponding to a combination of her distributivity and dependent choice axioms. We will also see that in the particular case κ = ω, our result is just Makkai and Reyes result for separable toposes, since the exactness property T in a separable topos is always satisfied. 1.1 κ-geometric logic Let κ be a regular cardinal. The syntax of κ-geometric logic consists of a (well-ordered) set of sorts and a set of function and relation symbols, these latter together with the corresponding type, which is a subset with less than κ many sorts. Therefore, we assume that our signature may contain relation and function symbols on γ < κ many variables, and we suppose there is a supply of κ many fresh variables of each sort. Terms and atomic formulas are defined as usual, and general formulas are defined inductively according to the following: Definition 1.1.1. If φ, ψ, {φα : α < γ} (for each γ < κ) and {ψα : α < δ} (for each δ) are κ-geometric formulas, the following are also formulas: α<γ φα, ∃α<γxαφ (also written ∃xγφ if xγ = {xα : α < γ}) and ψα, this latter provided that α<δ V ∪α<δFV (ψα), the set of free variables of all ψα, has cardinality less than κ. W We use sequent style calculus to formulate the axioms of first-order logic, as can be found, e.g., in [Joh02], D1.3. The system for κ-geometric logic is described in the following: Definition 1.1.2. The system of axioms and rules for κ-geometric logic consists of 1. Structural rules: (a) Identity axiom: φ ⊢x φ (b) Substitution rule: φ ⊢x ψ φ[s/x] ⊢y ψ[s/x] where y is a string of variables including all variables occurring in the string of terms s. (c) Cut rule: φ ⊢x ψ ψ ⊢x θ φ ⊢x θ 2. Equality axioms: 3 (a) ⊤⊢x x = x (b) (x = y) ∧ φ ⊢z φ[y/x] where x, y are contexts of the same length and type and z is any context containing x, y and the free variables of φ. 3. Conjunction axioms and rules: φi ⊢x φj i<γ ^ {φ ⊢x ψi}i<γ φ ⊢x ψi i<γ ^ for each cardinal γ < κ. 4. Disjunction axioms and rules: φj ⊢x φi i<γ _ {φi ⊢x θ}i<γ φi ⊢x θ i<γ _ for each cardinal γ. 5. Existential rule: φ ⊢xy ψ ========= ∃yφ ⊢x ψ where no variable in y is free in ψ. 6. Small distributivity axiom φ ∧ ψi ⊢x φ ∧ ψi i<γ i<γ _ _ for each cardinal γ. 4 7. Frobenius axiom: φ ∧∃yψ ⊢x ∃y(φ ∧ ψ) where no variable in y is in the context x. 8. Rule T : β φf ⊢yf ∃xgφg β <κ,f ∈ γ g∈γβ+1,g| =f _ β β φf ⊣⊢yf φf|α β < κ, limit β,f ∈ γ α<β^ φ∅ ⊢y∅ ∃β<δf xf|β+1 φf|β+1 f_∈B β<δ^f for each cardinal γ, where yf is the canonical context of φf , provided that, for β+1 every f ∈ γ , FV (φf ) = FV (φf|β ) ∪ xf and xf|β+1 ∩ FV (φf|β ) = ∅ for any <κ β < γ, as well as FV (φf )= α<β FV (φf|α ) for limit β.